Even the CVA-01 would have required a substantial redesign. Her plans were for an 18' height hanger deck. The Hawkeye is almost 19' tall to the top of the radar fairing. That would require a change of at least four feet in hanger deck height (Nimitz class carriers have a 26.5" height, Forrestal/Kitty Hawk was 25'). The Midway had a lot of trouble operating the Hawkeye due to part of her hanger having a 17.5' height restriction.
Yes, no worries if CVA01 & 02 get built. Grumman will just have to make some mods to lower the E2 by about 3 or 4 inches, or perhaps the British could build a trolley that holds an E2 with its wheel up for hangar maintenance. The rest of the time it would have to spend on the deck due to the hassle, probably a bit more than the rest of the squadrons, but nothing that couldn't be operated around I wouldn't think.
After the cancelation of the Nimrod A.E.W.3 the R.N is given responsibility for all maritime patrol and over ocean A.E.W duties and selects the aircraft designed for just that role.